Non-standard clock synchronization

Selleri proposes the following transformation equations
x=g(x'-vt')
t=t'/g
g=gamma.
Einstein's clock synchronization requires
t(E)=t(e)+x/c
t(E) reading of a clock located at a distance x from the origin, t(e) reading of a clock located at the origin when the synchronizing light signal is emitted,
Is there a link between the proposed transformation equations and the synchronization procedure?
Thanks for your answer

Selleri proposes the following transformation equations
x=g(x'-vt')
t=t'/g
g=gamma.
Einstein's clock synchronization requires
t(E)=t(e)+x/c
t(E) reading of a clock located at a distance x from the origin, t(e) reading of a clock located at the origin when the synchronizing light signal is emitted,
Is there a link between the proposed transformation equations and the synchronization procedure?
Thanks for your answer

DrGreg has already answered this question in a very beautiful way here.

Yes (except that the roles of "primed" and "unprimed" have been reversed). "Tangherlini Transform", "Selleri Transform", "Generalised Galilean Transform" (GGT), all refer to the same thing (though different authors attribute different significances to them). They also sometimes appear under the heading of "Lorentz Ether Theory" (LET), although Lorentz never used them.

Then use the Lorentz transform (and its inverse) for Einstein-observer I'E (x'E, t'E)

x'E = g(x - vt)t = g(t'E + vx'E/c2)

and you will obtain

x'T = x'Et'T = t'E + vx'E/c2

Note that c must be interpreted as the speed of light in the I frame, or equivalently the "two-way speed of light" in the I'T frame, because, in that frame the "one-way speed of light" depends on which direction you measure it in.

Yes (except that the roles of "primed" and "unprimed" have been reversed). "Tangherlini Transform", "Selleri Transform", "Generalised Galilean Transform" (GGT), all refer to the same thing (though different authors attribute different significances to them). They also sometimes appear under the heading of "Lorentz Ether Theory" (LET), although Lorentz never used them.

Note that c must be interpreted as the speed of light in the I frame, or equivalently the "two-way speed of light" in the I'T frame, because, in that frame the "one-way speed of light" depends on which direction you measure it in.

Thank you, DrGreg

I am not familiar with the "one-way speed" dependency on the direction of measurement. Does this follow from the definition of the metric in the Tangherlini/Selleri theory? Can you elaborate?

I am not familiar with the "one-way speed" dependency on the direction of measurement. Does this follow from the definition of the metric in the Tangherlini/Selleri theory? Can you elaborate?

Consider the equation x'E = ct'E. That describes something moving at the speed of light measured in the I'E frame. Now convert to the I'T frame. You get

x'T = ct'E

t'T = t'E + vt'E/c = (1 + v/c)t'E

From which the Tangherlini-speed of light is c / (1 + v/c).

Similarly, using x'E = -ct'E, the speed in the other direction is c / (1 - v/c).

But the there-and-back (2-way) average speed of light is still c.

You could also deduce this from the metric I quoted here, by solving ds = 0.

In fact, whenever the metric contains "off-diagonal" terms which "mix space and time", like dxdt, the speed of light, expressed in those coordinates, cannot be isotropic (=the same in all directions). (Think about solving ds = 0 and the roots of quadratic equations.)

Consider the equation x'E = ct'E. That describes something moving at the speed of light measured in the I'E frame. Now convert to the I'T frame. You get

x'T = ct'E

t'T = t'E + vt'E/c = (1 + v/c)t'E

From which the Tangherlini-speed of light is c / (1 + v/c).

Similarly, using x'E = -ct'E, the speed in the other direction is c / (1 - v/c).

But the there-and-back (2-way) average speed of light is still c.

You could also deduce this from the metric I quoted here, by solving ds = 0.

In fact, whenever the metric contains "off-diagonal" terms which "mix space and time", like dxdt, the speed of light, expressed in those coordinates, cannot be isotropic (=the same in all directions). (Think about solving ds = 0 and the roots of quadratic equations.)

Thank you, this is what I thought :

[tex]x=g(x'-vt')[/tex]
[tex]t=t'/g[/tex]

[tex]0=x^2-(ct)^2=g^2(x'-vt')^2-(ct'/g)^2[/tex]

Solving
[tex]g^2(x'-vt')^2-(ct'/g)^2=0[/tex] for [tex]\frac{x'}{t'}[/tex] I get the anysotropic speed.

You are right, the presence of the mixed term in x't' results immediately in anysotropic light speed.
Thank you so much for the incredible stuff, you are awesome!

Selleri proposes the following transformation equations
x=g(x'-vt')
t=t'/g
g=gamma.
Einstein's clock synchronization requires
t(E)=t(e)+x/c
t(E) reading of a clock located at a distance x from the origin, t(e) reading of a clock located at the origin when the synchronizing light signal is emitted,
Is there a link between the proposed transformation equations and the synchronization procedure?
Thanks for your answer

Sorry, I just realised I never gave an explicit answer to this question.

However, I've just noticed your descriptions of t(E) and t(e) in that post are confusing and maybe not quite what I am assuming. To avoid all doubt, I am interpreting:

t(e) to be the time, on the observer's own clock, when a light signal is emitted by the observer at the spatial origin.

t(E) to be the time, on an auxiliary clock located at a constant distance x from the observer, when the light signal is received.

The auxiliary clock is synchronised to make the equation (1) true.

For a given choice of c+ (between 0.5 and infinity), the set of all possible auxiliary clocks defines a (t,x) coordinate system.

Conclusion

Put the above two posts together to get my result.

The reason I originally put |x| instead of x is because of a technicality I haven't raised yet.

If you define your synchronisation operationally via light from the observer, then equation (1) would hold only for positive x. For negative x you'd need to replace x by -x. That's assuming that c+ is the 1-way speed of light outward away from the observer.

However, you might instead want c+ to be the 1-way speed of light in the positive x direction. In this case, equation (1) is correct even for negative x, but then the light is really travelling from the auxiliary clock to the observer (still in the positive x direction). In the case of Selleri coords, this second interpretation is the correct one, so I was wrong to use |x| in my previous post.

From a practical point of view, for negative x you can use the alternative equation